The No Boundary Proposal and i-Time

Skydive Phil and co recently released a couple of new videos in their Before the Big Bang series.  They have upped their tempo a little, with two episodes being released in less than four months.

I’ve not yet had time to look at “BtBB6: Can the Universe Create Itself?” (to which the answer would initially seem to be no, but I’ll reserve further comment until I’ve watched it).  I have, however, enjoyed “BtBB5: The No Boundary Proposal” and I must congratulate Phil on getting such luminaries together – including Stephen Hawking.

I do have a vested interest though, or two.

The first is that the model aligns very closely with something that I’ve pondered myself, which I’ll go into in a moment.  The second is that the No Boundary (or Hawking-Hartle) Proposal rips a hole in two of WLC’s favourite arguments, namely the one when he trundles out the BGV theorem and uses it to argue for his god and the one where he argues that the universe is contingent (while excluding necessity as underlying the beginning of the universe).

I’ll simply throw that out there and encourage the reader to watch the video.  Just keep in mind that if there is no conceptual issue with a natural universe that is not past eternal, then the BGV argument is short circuited and that if a natural universe can be necessary by virtue of physics, then the argument from contingency fails.  Or, at the very least, WLC and his ilk are constrained to a form of argument from incredulity (and/or from personal ignorance).

Now, the physics of the No Boundary Proposal.  I will say up front that I never considered an initial 4-space (ie a four-dimensional manifold) somehow transforming into a 3-space plus time, but I certainly have considered i-time.  Let me start from somewhere close to the beginning, or perhaps a long way before the beginning.  Bear with me if it seems like I am taking a weird detour.

One of the characteristics of a black hole is its density.  Density is simply the mass of something divided by its volume, so a very heavy (or, more correctly, very massive) thing in a small volume has high density, and something with mass spread out over a large volume has low density.  Lead is dense, air is not.

We tend to think of stars as very dense, and they certainly have a dense core, while black holes are thought of as either collapsed stars or, in the case of a supermassive black hole, many stars crushed together so tightly that nothing emerges from them – not even light.  So you'd be forgiven for thinking that black holes are very dense.  Not so fast ...

A star has a fuzzy boundary, so the answer to how dense it is is not as simple as with something more distinct like, say, a lump of granite.  Our sun has a core with a density in the order of 150 g/cm³ (liquid water is about 1 kg/litre, or 1 g/cm³) but if you measure from the “surface” of the sun, the photosphere, the average density is only 1.4 g/cm³ which is one quarter the density of the Earth at 5.51 g/cm³.  But at the photosphere, the sun has a density of 0.0000002 g/cm³ compared to the density of the atmosphere at sea level of 0.001225 g/cm³.  Only above 60km above sea level do you start to see atmospheric density in the order of the density of the sun at its "surface" (and then there's another 40km further to go until you are officially in space).

The sun doesn't really stop at the photosphere though, because there is also the corona, which extends out millions of kilometres from the sun (and is actually hotter than the "surface" of sun, up to 450 times hotter).  Note that the radius of the sun is a little under 700,000km, so the corona covers a much greater volume.  Then there is also the area in which solar wind is a factor, which could be argued as being the heliosheath and is sort of equivalent to the atmosphere of a planet (it's a "protective" bubble against the interstellar medium).  This heliosheath extends far out beyond Pluto or about 120 au (astronomical units, so 120 times further out than Earth).

Unlike a star, a black hole has a very well-defined boundary, known as the event horizon which itself can be defined by the Schwarzschild radius or the distance from the centre of the mass of a (non-rotating, spherically symmetric) black hole at which a photon can only just escape (because the escape velocity is the speed of light).  The Schwarzschild radius is directly proportional to the mass (M*2G/c²).  The volume defined by a radius is, however, proportional to the inverse of the radius cubed (4/3.π/r³).  The upshot is that the density of black hole is proportional to the inverse square of the mass – as the mass increases, the density of the black hole goes down (proportional to the square of the increase of the mass).

If you plug in the mass of the universe into the equation (3c⁶/(32G³.M²), you get a density in the order of the density of the universe – implying that we may be in a black hole.

(Note: if the black hole in question is rotating or not spherically symmetric, the argument that we are inside a black hole only improves.)

If we are indeed inside a black hole, the question then arises as to what is outside the black hole.

Let’s leave that question aside for a moment and consider instead what happens to time as you approach the Schwarzschild radius (or event horizon) of a black hole.  The equation for this is given by t_o = t_f.√(1-r_s/r) which basically tells us that, as your distance from the centre of the black hole (r) approaches the Schwarzschild radius (r_s), the time you experience (t_o) decreases as compared to that of a distant observer (t_f) – or time slows down as you approach the event horizon.  This is a phenomenon referred to in the movie Interstellar where travellers deep in the gravity well of a very heavy (massive) planet left their buddy up in space alone for ages while they only experienced a relatively short time on the (liquid) surface.

But this is all about what happens outside the black hole, on the “safe” side of the event horizon at r_s.  If you look at that equation again, you’ll see that something funky happens with time when r < r_s.  You get the square root of a negative number, which is imaginary time, or i-time.  Inside a black hole, time shoots off orthogonally from time outside the black hole, and it does so with no limit because as r->0, r_s/r->infinity (and (1-r_s/r)-> negative infinity and √(1-r_s/r)->i-infinity).

There’s a spatial effect as well.  Compared to a distant observer, rulers in a gravitational well (as in close to a black hole) shrink and once, inside past event horizon, they expand out into i-space.  In other words, within a black hole, there is effectively infinite space and infinite time, but it’s all orthogonal to what is outside.

Now consider the view from inside a black hole.  The entire history of the universe outside will effectively happen instantaneously at t=0, everything that is ever sucked into the black hole will appear instantaneously, having been ripped apart by the transition through the event horizon, with no information remaining in it. There will, be at t=0, basically no space (there’s a limit to how little space there is due to how much mass-energy can fit into it) but as space expands out (orthogonally to space outside the black hole) you basically end up with a Big Bang.  There is a natural correlation between a t=0 and a severely limited amount of space, there’s also a natural arrow of time, and a natural expansion.

So that’s basically how I think of the Big Bang, with each universe eventually (after an eternity) ending up ripped apart inside a new black hole which has a new Big Bang and the routine keeps going, with multiple eternities (and maybe even multiply infinite space, although that is less certain to me, particularly given the initial non-zero space condition).  The thing that I find so interesting with the No Boundary Proposal is not so much that it parallels my model (which it only does to some small extent), but rather that there is a willingness to consider what I call i-time especially given that it is an unavoidable consequence of my model.  There’s also the fact that, as far as we inside our black hole are concerned, the universe outside is as good as a four-dimensional manifold – what happens (or rather happened) out there is all done and dusted, set in stone in a way because it’s all in the past from our perspective (and it all happened instantaneously at t=0) and, from our perspective, the entire history of that external universe is in imaginary time, since that time is orthogonal to our time.

The benefit of my model is there’s a clear explanation as to the transition to our time and our space.  The down side is, well, there are truly brilliant brains looking at the maths behind the four-dimensional manifold and they seem to think that it works.  There’s just my brain looking at my model.